A link function specification test in the single functional index model

In this paper a test for specification in functional regression with scalar response that exploits semi-parametric principles is illustrated. Once the test statistics is defined, its asymptotic null distribution is derived under suitable conditions. The finite sample performances of the test are analyzed through a simulation study by using both the asymptotic p-value and some bootstrap approaches. To appreciate the potentialities of the method, an application to a spectrometric real dataset is performed.